mathematics
Article Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem
Ruslan Gabdullin 1, Vladimir Makarenko 1 and Irina Shevtsova 1,2,3,4,*
1 Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected] (R.G.); [email protected] (V.M.) 2 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China 3 Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia 4 Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia * Correspondence: [email protected]
Abstract: Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones. Keywords: central limit theorem; Lindeberg’s theorem; normal approximation; asymptotically exact Citation: Gabdullin, R.; Makarenko, constant; asymptotically best constant; uniform distance; Lindeberg fraction; truncated moment; V.; Shevtsova, I. Asymptotically Exact absolute constant Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem. Mathematics 2021, 9, 501. https://doi.org/10.3390/math9050501 1. Introduction In various applications of probability theory, one has to approximate an unknown Academic Editor: Antonio Di distribution of a sum of independent random variables with some known law. Such prob- Crescenzo lems arise, for example, in insurance, financial mathematics, reliability theory, queueing theory, and many other areas. The most common approximation is the normal one which Received: 6 February 2021 is based on the central limit theorem. The adequacy of the normal approximation can be Accepted: 24 February 2021 Published: 1 March 2021 estimated with the help of convergence rate estimates in the central limit theorem such as the celebrated Berry–Esseen [1,2] inequality (in terms of full moments and under the addi-
Publisher’s Note: MDPI stays neutral tional moment-type assumptions), or Osipov–Petrov’s [3,4], Esseen’s [5], Rozovskii’s [6], with regard to jurisdictional claims in Wang-Ahmad’s [7] inequalities and their generalizations [8–11] (in terms of truncated published maps and institutional affil- moments without any additional assumptions). However, the most natural estimates, such iations. as Esseen’s, Rozovskii’s and Wang–Ahmad’s inequalities contained unknown constants, and their application in practice was made possible only by the results of [8,11], where in particular, the unknown constants in the above inequalities were evaluated. A detailed overview of the cited inequalities can be found in [11] and for brevity, we do not duplicate it here. Since the crucial role in estimation of the adequacy of the normal approximation Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. is played by the values (upper bounds, in fact) of appearing absolute constants, it is very This article is an open access article important to understand how accurate the existing upper bounds for the constants are, distributed under the terms and how much they might be lowered and if it is worth trying to improve the method of their conditions of the Creative Commons evaluation. The problem becomes much deeper as soon as we observe that estimates of the Attribution (CC BY) license (https:// accuracy of the normal approximation are usually used with large sample sizes or when creativecommons.org/licenses/by/ the majorizing expressions are assumed to be small, so that, in fact, not only the absolute 4.0/). values of the appearing constants are of interest, but also their presumably more optimistic
Mathematics 2021, 9, 501. https://doi.org/10.3390/math9050501 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 501 2 of 32
(smaller) values which may be used under the corresponding asymptotic assumptions. Every set of asymptotic assumptions generates the corresponding asymptotic constant. Hence, we can introduce a whole classification of asymptotic constants. The present study is devoted to investigation of the various asymptotic constants and the main purpose is to construct their lower bounds. Let X1, X2, ... , Xn be independent random variables (r.v.’) with distribution functions 2 (d.f.’s) Fk(x) = P(Xk < x), x ∈ R, expectations EXk = 0, variances σk = VarXk, k = 1, ... , n, and such that n 2 2 Bn := ∑ σk > 0. k=1 For n = 1, 2, . . . denote
n Sn − ESn Xk Sn = X1 + X2 + ··· + Xn, Sen = √ = ∑ , VarSn k=1 Bn Z x 1 −t2/2 Φ(x) = √ e dt, x ∈ R, ∆n = ∆n(F1,..., Fn) = sup P(Sen < x) − Φ(x) , 2π −∞ x∈R 2 2 3 σk (z) = EXk 1(|Xk| > z), µk(z) = EXk 1(|Xk| < z), k = 1, . . . , n 1 n 1 n ( ) = ( ) = E 3 (| | < ) Mn z : 3 ∑ µk zBn 3 ∑ Xk 1 Xk zBn , Bn k=1 Bn k=1 1 n ( ) = E| |3 (| | < ) Λn z : 3 ∑ Xk 1 Xk zBn , Bn k=1 1 n 1 n ( ) = 2( ) = E 2 (| | ) > Ln z 2 ∑ σk zBn 2 ∑ Xk 1 Xk > zBn , z 0. Bn k=1 Bn k=1
The function Ln( · ) is called the Lindeberg fraction. It is easy to see that |Mn(z)| 6 Λn(z), z > 0. In case of independent identically distributed (i.i.d.) r.v.’s X1, ... , Xn we denote their common d.f. by F and write ∆n(F) := ∆n(F,..., F). In [8] it was proved that
∆n 6 AE(ε, γ) sup {γ|Mn(z)| + zLn(z)}, (1) 0 where the functions AE(ε, γ), AR(ε, γ) depend only on ε and γ (that is, they turn into absolute constants as soon as ε and γ are fixed), both are monotonically non-increasing with respect to γ > 0, and AE(ε, γ) is also non-increasing with respect to ε > 0. The question on the boundedness of AR(ε, γ) as ε → ∞ is still open, while AE(0+, γ) = AR(0+, γ) = ∞ for every γ > 0 and AE(ε, 0+) = AR(ε, 0+) = ∞ for every ε > 0. To avoid ambiguity, in what follows by constants appearing here in various inequalities we mean their exact values; in particular, in majorizing expressions — their least possible values. Upper bounds for the constants AE(ε, γ) and AR(ε, γ) for some ε and γ computed in [8] are presented in Tables1 and2, respectively. Here the symbol γ∗ stands for the point of minimum of the upper bound for AR(ε, γ), obtained within the framework of the method used in [8], i.e., the upper bound for AR(ε, γ) found in [8] remains constant as γ > γ∗ grows for every fixed ε > 0. More precisely, the quantity γ∗ is defined as follows: √ √ 2 4 1/4 γ∗ = 1/ 6κ = (1 − t + t ) /(t 3) = 0.5599 . . . , Mathematics 2021, 9, 501 3 of 32 where t ∈ (π/2, π) is the unique root of the equation tan t = t/(1 − t2), q −2 2 2 2 κ = x (cos x − 1 + x /2) + (sin x − x) = 0.5315 . . . , x=x0 x0 = 5.487414 . . . is the unique root of the equation 8(cos x − 1) + 8x sin x − 4x2 cos x − x3 sin x = 0, x ∈ (π, 2π). Table 1. Two-sided bounds for the constants AE(ε, γ) from inequality (1), γ∗ = 0.5599 ... Upper bounds were obtained in [8], and the lower ones in Theorem3 below. ε γ AE(ε, γ) 6 AE(ε, γ) > ε γ AE(ε, γ) 6 AE(ε, γ) > 1.21 0.2 2.8904 0.5006 ∞ 1 2.6588 0.3703 1.24 0.2 2.8900 0.4889 ∞ 0.97 2.6599 0.3736 ∞ 0.2 2.8846 0.4876 2.56 ∞ 2.6500 0 1.76 0.4 2.7360 0.4606 2.62 5 2.6500 0.2126 5.94 0.4 2.7300 0.4606 2.65 4 2.6500 0.2117 ∞ 0.4 2.7299 0.4606 2.74 3 2.6500 0.2458 1 γ∗ 2.7367 0.5795 3.13 2 2.6500 0.3018 1.87 γ∗ 2.6999 0.4359 4 1.62 2.6500 0.3222 ∞ γ∗ 2.6919 0.4359 5.37 1.5 2.6500 0.3294 1 0.72 2.7298 0.5746 ∞ 1.43 2.6500 0.3339 1 ∞ 2.7286 0 ∞ ∞ 2.6409 0 4.35 1 2.6600 0.3703 0+ ∀ ∞ ∞ Table 2. Two-sided bounds for the constants AR(ε, γ) from inequality (2), γ∗ = 0.5599 ... Upper bounds were obtained in [8], and the lower ones in Theorem3 below. ε γ AR(ε, γ) 6 AR(ε, γ) > ε γ AR(ε, γ) 6 AR(ε, γ) > 1.21 0.2 2.8700 0.5048 1.99 γ∗ 2.6600 0.4357 5.39 0.2 2.8635 0.5000 2.12 γ∗ 2.6593 0.4397 1.76 0.4 2.6999 0.4485 3 γ∗ 2.6769 0.4586 2.63 0.4 2.6933 0.4675 5 γ∗ 2.7562 0.4784 0.5 γ∗ 3.0396 1.1329 0+ ∀ ∞ ∞ 1 γ∗ 2.7286 0.5795 In the same paper [8] there were found sharpened upper bounds for the constants AE(ε, γ) and AR(ε, γ) provided that the corresponding fractions LE,n(ε, γ) := sup {γ|Mn(z)| + zLn(z)}, LR,n(ε, γ) := γ|Mn(ε)| + sup zLn(z). 0 take small values. In particular, there were introduced asymptotically exact constants ∆ (F ,..., F ) AAE(ε, γ) := lim sup sup n 1 n : L (ε, γ) = ` , ε, γ > 0, (3) E ` E,n `→0 n,F1,...,Fn ∆ (F ,..., F ) AAE(ε, γ) := lim sup sup n 1 n : L (ε, γ) = ` , ε, γ > 0, (4) R ` R,n `→0 n,F1,...,Fn Mathematics 2021, 9, 501 4 of 32 and the following upper bounds were obtained for them: t2 t2 p 2 t2 AE 4 1 κ γ ε γ 2(6κγ + 1) 3 γ AE (ε, γ) 6 √ + Υ 1, + Υ 2, + Γ , , (5) 2π π ε 2ε2 12 2ε2 6γ 2 2ε2 t2 t2 t2 AE 4 1 h κ 1,γ ε 1,γ ε 2,γ AR (ε, γ) 6 √ + Υ 1, + Υ 2, + Γ 2, + 2π π ε 2ε2 12 2ε2 6 2ε2 √ √ 2 2 2 π 3 t1,γ 3 t2,γ i + − Υ , − Γ , , (6) 6γ 2 2 2ε2 2 2ε2 R ∞ r−1 −t R x r−1 −t where Γ(r, x) := x t e dt, Υ(r, x) := 0 t e dt = Γ(r) − Γ(r, x), r, x > 0, are the upper and the lower gamma-functions, respectively, q q 2 2 −1 −1 2 tγ := γ (γ/γ∗) + 1 − 1 , t2,γ = 2 max γ , γ∗ , t1,γ := t2,γ 1 − (1 − (γ/γ∗) )+ , κ = 0.5315 ..., γ∗ = 0.5599 ... were defined above. The values of the upper bounds for AE AE the asymptotically exact constants AE (ε, γ) and AR (ε, γ) in (5) and (6) for some ε > 0 and γ > 0 are given in the third and the seventh columns of Table3, respectively. AE AE AE Moreover, in [8] it was shown that the asymptotically exact constants A• ∈ {AE , AR } are unbounded as γ → 0: AE ∆n(F) A• (ε, γ) > sup lim sup → ∞, γ → 0, ∀ε > 0. F n→∞ L•,n(ε, γ) Let us note that estimate (2) with ε = γ = 1 coincides with the Rozovskii inequal- ity [6] ([Corollary 1]) and establishes an upper bound for the appearing absolute constant AR(1, 1) 6 AR(1, γ∗) 6 2.73. Estimate (1) with ε = γ = 1 and ε → ∞, γ = 1 improves both Esseen’s inequalities from [5], where the absolute value sign and the least upper bound with respect to z ∈ (0, ε) stand inside the sum in comparison with (1). In partic- ular, estimate (1) yields upper bounds for the absolute constants in Esseen’s inequalities AE(1, 1) 6 AE(1, 0.72) 6 2.73 and AE(∞, 1) 6 AE(∞, 0.97) 6 2.66 which were remaining unknown for a long time. Esseen’s inequality where the least upper bound is taken over a bounded range (see (1) with ε = γ = 1) yields, in its turn, the Osipov inequality [3]: ∆n 6 AE(1, 1) sup {|Mn(z)| + zLn(z)} 6 AE(1, 1) sup {Λn(z) + zLn(z)} = 0 = AE(1, 1)(Λn(1) + Ln(1)) = AE(1, 1) inf(Λn(ε) + Ln(ε)) (7) ε>0 (for details see [8]). The latest bound obtained by Osipov [3] with some constant AE(1, 1) (whose best known value 1.87 is published in [12]) yields, in its turn, the Lindeberg theorem: indeed, under the Lindeberg condition sup lim L (ε) = 0 and with the account → n ε>0 n ∞ of Λn(ε) 6 ε, from (7) we have ∆n 6 AE(1, 1) inf(ε + Ln(ε)) → 0, n → ∞. ε>0 Hence, by Feller’s theorem, in case of uniformly infinitesimal random summands (in partic- ular, in the i.i.d. case) the right- and the left-hand sides of (7) are either both infinitesimal or both do not tend to zero. According to the terminology, introduced by Zolotarev [13], such convergence rate estimates are called natural. Together with (1), inequality (2) is a natural convergence rate estimate in the Lindeberg–Feller theorem certainly under the additional assumption of existence of such an ε0 > 0 that Mn(ε0) = 0 for all sufficiently large n (see, e.g., [11]), in particular, if the r.v.’s X1,..., Xn have symmetric distributions. Mathematics 2021, 9, 501 5 of 32 AE AE Table 3. Values of the two-sided bounds of the asymptotically exact constants AE (ε, γ) and AR (ε, γ) in (5) and (6) for some ε > 0 and γ > 0. Recall that γ∗ = 0.5599 ... . Upper bounds were obtained in [8], and the lower ones in Corollary1 below. AE AE AE AE ε γ AE (ε, γ) 6 AE (ε, γ) > ε γ AR (ε, γ) 6 AR (ε, γ) > 0.6 0.3 1.9225 0.5251 1.21 0.2 1.9348 0.5013 1.21 0.2 1.9546 0.5384 1.89 0.2 1.9300 0.5013 2.06 0.2 1.9500 0.5384 2.77 0.2 1.9289 0.5013 ∞ 0.2 1.9488 0.5384 5.39 0.2 1.9584 0.5013 1.48 0.4 1.8100 0.5111 1.41 0.4 1.7798 0.4516 ∞ 0.4 1.8001 0.5111 1.76 0.4 1.7725 0.4516 1.89 γ∗ 1.7714 0.4868 1.99 0.4 1.7713 0.4516 2.03 γ∗ 1.7700 0.4868 2.63 0.4 1.7785 0.4516 ∞ γ∗ 1.7638 0.4868 0.5 γ∗ 1.9475 0.3329 1 γ∗ 1.8060 0.4868 1 γ∗ 1.7916 0.4184 1 0.67 1.7997 0.4685 1.52 γ∗ 1.7500 0.4184 1 ∞ 1.7915 0.1994 1.89 γ∗ 1.7439 0.4184 2.24 1 1.7400 0.4097 1.99 γ∗ 1.7442 0.4184 ∞ 1 1.7319 0.4097 2.12 γ∗ 1.7455 0.4184 3.07 ∞ 1.7200 0.1994 3 γ∗ 1.7710 0.4184 3.2 5 1.7200 0.1994 5 γ∗ 1.8650 0.4184 3.28 4 1.7200 0.2240 4 2.4 1.7200 0.3059 5 2.06 1.7200 0.3284 5.37 2 1.7200 0.3324 ∞ 1.83 1.7200 0.3435 ∞ ∞ 1.7146 0.1994 Let us also note that Esseen-type inequality (1) not only links the criteria of convergence with the rate of convergence, as Osipov’s inequality does, but also provides a numerical demonstration of the Ibragimov’s criteria [14] of the rate of convergence in the CLT to be −1/2 −1/2 of order order O(n ). According to [14], in the i.i.d. case we have ∆n = O(n ) as 2 n → ∞ if and only if max{|µ1(z)|, zσ1 (z)} = O(1) as z → ∞. Inequality (1) trivially yields −1/2 the sufficiency of the Ibragimov condition to ∆n = O(n ) as n → ∞. Let us denote by G a set of all non-decreasing functions g : [0, ∞) → [0, ∞) such that g(z) > 0 for z > 0 and z/g(z) is also non-decreasing for z > 0. The set G was initially introduced by Katz [15] and used later in the works [4,9–12,16]. In [9] it was proved that (i) For every function g ∈ G and a > 0 n z o g(z) n z o g (z, a) := min , 1 max , 1 := g (z, a), z > 0, (8) 0 a 6 g(a) 6 a 1 with g0( · , a), g1( · , a) ∈ G. (ii) Every function from G is continuous on (0, ∞). Mathematics 2021, 9, 501 6 of 32 Property (8) means that every function from G is asymptotically (as its argument goes to infinity) between a constant and a linear function. For example, besides g0, g1, the class G also includes the following functions: δ gC (z) ≡ 1, g∗(z) = z, c · z , c · g(z), z > 0, for all c > 0, δ ∈ [0, 1] and g ∈ G. For g ∈ G we set n n 1 g(z) 2 LE n(g, ε, γ) = sup γ µ (z) + z σ (z) , B2 g(B ) z ∑ k ∑ k n n 0 n n 1 g(εBn) 2 LR n(g, ε, γ) = γ µ (εBn) + sup g(z) σ (z) , B2 g(B ) εB ∑ k ∑ k n n n k=1 0 Please note that the introduced fractions with g = g∗ coincide with the fractions in the Esseen- and Rozovskii-type inequalities (1), (2) considered above: LE,n(g∗, ε, γ) = LE,n(ε, γ), LR,n(g∗, ε, γ) = LR,n(ε, γ), ε, γ > 0. Inequalities (1) and (2) were generalized in [11] in the following way: ∆n 6 CE(ε, γ) · LE,n(g, ε, γ), (11) ∆n 6 CR(ε, γ) · LR,n(g, ε, γ), ε, γ > 0, g ∈ G, (12) where CE(ε, · ) = AE(ε, · ), ε ∈ (0, 1], in particular, CE(+0, · ) = ∞, CE(ε, · ) 6 AE(1, · ), ε > 1; CR(ε, · ) = AR(ε, · ), ε ∈ (0, 1], in particular, CR(+0, · ) = ∞, CR(ε, · ) 6 εAR(ε, · ), ε > 1, CR(∞, · ) = ∞ (recall that according to the above convention, all the equalities between the constants including CE(0, · ) = CR(0, · ) = CR(∞, · ) = ∞ are exact with formal definitions of CE(ε, γ) and CR(ε, γ) being given in (19) below). It is easy to see that the both fractions L• , n ∈ {LE,n, LR,n} are invariant with respect to scale transformations of g ∈ G: L• , n(cg, · , · ) = L• , n(g, · , · ), c > 0. Moreover, in [11] ([Theorem 2]) it was proved that for all ε, γ > 0 1 6 LE,n(g1, ε, γ) 6 max{ε, 1}· max{γ, 1}, (13) 1 6 LR,n(g1, ε, γ) 6 max{ε, 1}· (γ + 1). (14) Extreme properties of the functions g0(z) := Bng0(z, Bn) = min{z, Bn}, g1(z) := Bng1(z, Bn) = max{z, Bn}, z > 0, Mathematics 2021, 9, 501 7 of 32 in (8) with a := Bn yield L• , n(g0, · , · ) 6 L• , n(g, · , · ) 6 L• , n(g1, · , · ), g ∈ G, (15) for every fixed set of distributions of X1, ... , Xn, so that the extreme values of the constants CE and CR in (11) and (12) with fixed n and F1, ... , Fn are attained at g = g0. Moreover, with the extreme functions g the fraction L• , n ∈ {LE,n, LR,n} satisfy the following relations for ε 6 1: L• , n(g0, ε, · ) = L• , n(g∗, ε, · ) = L• , n(ε, · ), (16) L• , n(g1, ε, · ) = L• , n(gC, ε, · ). (17) Inequality (11) also generalizes and improves up to the values of the appearing absolute constant the classical Katz–Petrov inequality [4,15] (which is equivalent to the Osipov inequality [3]) because of involving the algebraic truncated third order moments instead of the absolutes ones and also a recent result of Wang and Ahmad [7] at the expense of moving the modulus and the least upper bound signs outside of the sum sign. In particular, inequality (11) established an upper bound of the constant in the Wang–Ahmad inequality CE(∞, 1) 6 AE(1, 1) 6 2.73. A detailed survey and analysis of the relationships between inequalities (1), (2), (11), (12) with inequalities of Katz [15], Petrov [4], Osipov [3], Esseen [5], Rozovskii [6], and Wang–Ahmad [7] can be found in papers [8,11]. The main goal of the present work is construction of the lower bounds of the absolute constants CE(ε, γ), CR(ε, γ) in inequalities (11), (12), and also of the constants AE(ε, γ), AR(ε, γ) in inequalities (1), (2), in particular, we show that even in the i.i.d. case CE(1, 1) = AE(1, 1) > 0.5685, CR(1, 1) = AR(1, 1) > 0.5685, CE(∞, 1) > 0.5685, AE(∞, 1) > 0.3703. We consider various statements of the problem of construction of the lower bounds, namely we introduce a detailed classification of the asymptotically exact constants and construct their lower bounds. As a corollary, we obtain two-sided bounds for the asymptotically AE AE exact constants AE (ε, γ) and AR (ε, γ) defined in (3) and (4), in particular, we show that √ 10 + 3 √ AE 0.4097 . . . = 6 AE (1, 1) 6 1.80. 6 2π 1 √ AE 0.3989 . . . = 6 AR (1, 1) 6 1.80. 2π The paper is organized as follows. In Section2 we introduce exact, asymptotically exact and asymptotically best constants defining the corresponding statements for the construction of the lower bounds. Sections4–6 are devoted namely to the construction of the lower bounds for the introduced constants. Section3 contains some auxiliary results which might represent an independent interest, in particular, the values of the fractions LE,n(g, ε, γ) and LE,n(g, ε, γ) are found for all n, ε, γ > 0 and some g ∈ G in the case where X1,..., Xn have identical two-point distribution. 2. Exact, Asymptotically Exact and Asymptotically Best Constants Following [13,17–25], let us define exact, asymptotically exact and asymptotically best constants in inequalities (11), (12). Let F be a set of all d.f.’s with zero means and finite second order moments. Denote ∆ (F ,..., F ) ∆ (F ,..., F ) C (g, ε, γ) = sup n 1 n , C (g, ε, γ) = sup n 1 n . (18) E L (g, ε, γ) R L (g, ε, γ) F1,...,Fn∈F, E,n F1,...,Fn∈F, R,n n∈N: Bn>0 n∈N: Bn>0 Mathematics 2021, 9, 501 8 of 32 Please note that the fractions LE,n(g, ε, γ), LR,n(g, ε, γ) also depend on d.f.’s F1, ... , Fn, but we omit these arguments for the sake of brevity. The constants CE(g, ε, γ) and CR(g, ε, γ) are the minimal possible (exact) values of the constants CE(ε, γ) and CR(ε, γ) in inequalities (11), (12) for the fixed function g ∈ G, while their universal values supg∈G CE(g, ε, γ), supg∈G CR(g, ε, γ), that provide the validity of the inequalities under consideration for all g ∈ G are called exact constants and namely they are the minimal possible (exact) values of the constants CE(ε, γ) and CR(ε, γ) in (11) and (12), respectively. In order not to introduce excess notation and following the above convention, we use namely these exact values for the definitions of CE(ε, γ) and CR(ε, γ) in the present work: CE(ε, γ) = sup CE(g, ε, γ), CR(ε, γ) = sup CR(g, ε, γ), (19) g∈G g∈G and call them the exact constants. Please note that AE(ε, γ) = CE(g∗, ε, γ) 6 CE(ε, γ), AR(ε, γ) = CR(g∗, ε, γ) 6 CR(ε, γ), ε, γ > 0, hence, every lower bound for the constants AE(ε, γ), AR(ε, γ) serves as a lower bound for the constants CE(ε, γ), CR(ε, γ) as well. The least upper bounds in (18) are taken without any restrictions on the values of the fractions LE,n(g, ε, γ), LR,n(g, ε, γ), while inequalities (11), (12) represent the most interest with small values of these fractions, when the normal approximation is adequate and only the concrete estimates of its accuracy are needed. That is why it is interesting to study not only the absolute, but also the asymptotic constants in (11), (12), some of which were already introduced in (3), (4). Since we are interested in the lower bounds, we assume that X1, ... , Xn have identical distributions. Generalization of the definitions introduced below to the non-i.id. case is not difficult (see, for example, definitions (3) and (4) of the asymptotically exact constants for the general case). For each of the fractions Ln = Ln(F, g, ε, γ) ∈ {LE,n, LR,n} appearing in inequalities (11), (12) we define the asymptotically best constants CAB(g, ε, γ) = sup lim sup ∆n(F)/Ln(F, g, ε, γ), (20) F∈F n→∞ the upper asymptotically exact constants CAE(g, ε, γ) = lim sup sup ∆n(F)/Ln(F, g, ε, γ), (21) n→∞ F∈F the asymptotically exact constants CAE(g, ε, γ) = lim sup sup ∆n(F)/`, (22) `→0 n∈N, F∈F : Ln(F,g,ε,γ)=` the lower asymptotically exact constants CAE(g, ε, γ) = lim sup lim sup sup ∆n(F)/`, (23) `→0 n→∞ F∈F : Ln(F,g,ε,γ)=` the conditional upper asymptotically exact constants ∗ CAE(g, ε, γ) = sup lim sup sup ∆n(F)/`. (24) `>0 n→∞ F∈F : Ln(F,g,ε,γ)=` In order not to introduce excess indexes, we use identical notation for the asymptotic constants in (11), (12), in what follows every time specifying the inequality in question. All the introduced constants aim to improve the structure of the simplest upper bounds of Mathematics 2021, 9, 501 9 of 32 the form ∆n 6 C(g, ε, γ)Ln(g, ε, γ) with the help of introducing an additional term o(Ln) which is allowed to be infinitesimal of a higher order than Ln as Ln → 0: ∆n 6 C∗(g, ε, γ)Ln(F, g, ε, γ) + o(Ln(F, g, ε, γ)), n → ∞, (25) where the assumptions on o(Ln) define the value of the corresponding minimal constant C∗. For example, in (20) it is supposed that the distribution of the random summands does not depend on the number of summands n and, hence, Ln → 0 if and only if n → ∞, so that o(Ln(F, g, ε, γ)) in (25) is not obliged to tend to zero uniformly for all d.f.’s F ∈ F as n → ∞. All the rest introduced constants allow a double array scheme. The values of Ln are obliged to be infinitesimal in (20), (22), and (23). The distinction between (22) and (23) is in the upper bound and the limit with respect to n, so that CAE(g, ε, γ) > CAE(g, ε, γ), where the strict inequality may also take place, as it happens indeed, for example, with the similar constants in the classical Berry–Esseen inequality [19–24]. In terms of inequalities, this means that inequality (25) with C∗ = CAE assumes that o(Ln(F, g, ε, γ)) tends to zero uniformly for all d.f.’s F ∈ F with fixed value of Ln(F, g, ε, γ) = ` as ` → 0, while taking C∗ = CAE in (25) leads to separating of o(Ln(F, g, ε, γ)) in (25) into two terms: sup ∆n = CAE(g, ε, γ)` + rn(`, g, ε, γ), F∈F : Ln(F,g,ε,γ)=` where lim sup lim sup |rn(`, g, ε, γ)| = 0. The constants in the classical Berry–Esseen in- `→0 n→∞ equality similar to those defined in (20)–(24) were first considered in [18] for (20),[17] for (22),[13,26] for (21),[23] for (23), and [25] for (24). The upper asymptotically ex- act (21) and the conditional upper asymptotically exact (24) constants are linked by the ∗ following relation CAE 6 CAE by definition, and we shall construct lower bounds namely ∗ for CAE. As for the constants CAE, we introduce them here to pay tribute to the classical works [13,26]. The function g in (20), (21), (24) may be arbitrary from the class G, while the constants CAB(g, ε, γ) and CAE(g, ε, γ) (see (22) and (23)) are defined not for all g ∈ G. For example, CAB(g1, ε, γ) and CAE(g1, ε, γ) are not defined in any of the inequalities (11), (12), since the corresponding fractions LE,n(g1, ε, γ), LR,n(g1, ε, γ) are bounded from below by one uniformly with respect to ε and γ (see (13) and (14)) and, hence, cannot be infinitesimal. Finally, note that definitions (20)–(24) immediately yield the relations ∗ ∗ max{CAE, CAB} 6 min{CAE, CAE}, max{CAE, CAE} 6 CAE (26) where we omitted the arguments g, ε, γ for clarity). 3. Two-Point Distributions Most of the lower bounds will be obtained by the choice of a two-point distribution r q r p P X = = 1 − P X = − = p, q = 1 − p ∈ (0, 1), k = 1, . . . , n, (27) k p k q for the random summands. The present section contains the corresponding required results. In particular, we will find values of the fractions LE,n(g, ε, γ), LR,n(g, ε, γ) with g = g∗, gC, g0, g1 for all ε, γ > 0 (Theorem1). We will also investigate the uniform distance between the d.f. of (27) and the standard normal d.f. Φ and find the corresponding extreme values of the argument of d.f.’s (see Theorem2): ( p Φ( p/q) − p, 0 < p < 1 , ( ) = |P( < ) − ( )| = 2 ∆1 p : sup X1 x Φ x p 1 (28) x∈R Φ( q/p) − q, 2 6 p < 1. Mathematics 2021, 9, 501 10 of 32 3.1. Computation of the Fractions 2 2 For distribution (27) we have EXk = 0, EXk = 1, k = 1, . . . , n, Bn = n, √ √ g0(z) = min{z, n}, g1(z) = max{z, n}, g∗(z) = z, gC (z) = 1, z > 0. Recall that for ε 6 1 LE,n(g0, ε, ·) = LE,n(g∗, ε, ·) = LE,n(ε, ·), LE,n(g1, ε, ·) = LE,n(gC, ε, ·), LR,n(g0, ε, ·) = LR,n(g∗, ε, ·) = LR,n(ε, ·), LR,n(g1, ε, ·) = LR,n(gC, ε, ·). That is why in the formulation of the next Theorem1, we do not indicate values of LE,n(g0, ε, ·), LE,n(g1, ε, ·), LR,n(g0, ε, ·), LR,n(g1, ε, ·) for ε 6 1 separately. Theorem 1. (i) For all n ∈ N, p ∈ [1/2, 1) and ε, γ > 0 we have ε, nε2 q , 6 p q 3 q q q q 2 p LE,n(g∗, ε, γ) = max np , γ np + εp , p < nε 6 q , (29) √ 2 2 2 p max q, (γq + p )1(p > 1/2), γ(p − q) / npq, nε > q , LE,n(g0, ε, γ) = LE,n(g∗, ε ∧ 1, γ), 2 q 1, nε 6 p , + q < 2 p LE,n(gC, ε, γ) = max{1, γq p}, p nε 6 q , (30) n o 1 γ(p−q) 2 p max 1, (γq + p)1(p > 2 ), p , nε > q , 2 q ε, nε 6 p , q 3 nq o γ q + max q , εp , q < nε2 p , LR,n(g∗, ε, γ) = np np p 6 q (31) q q 3 γ√(p−q) + q p 2 > p npq max np , nq , nε q , 2 q 1, nε 6 p , q ( ) = γ q3 q 2 p LR,n gC, ε, γ ε np + 1, p < nε 6 q , (32) γ√(p−q) 2 p ε npq + 1, nε > q , (ii) For all ε > 1, γ > 0, n ∈ N LE,n(g1, ε, γ) = 1, if p = 1/2, 3/2 max 1, γq + p, γ√q + pε , nε2 p , np 6 q 2 2 γ√q +p γ√(p−q) p 2 LE,n(g1, ε, γ) = max 1, γq + p, , , n 6 < nε , if p > 1/2. npq npq q γ(p−q) p max 1, γq + p, p , n > q , (iii) For all ε > 1, γ > 0, n ∈ N, and p ∈ [1/2, 1) q q3 q p γ + 1, n < nε2 , ε np 6 p 6 q γ(p−q) q p √ + 1, n 6 6 < nε2, ε npq p q γ q q3 nq q o q p + max , p , < n < nε2 , LR,n(g0, ε, γ) = ε np np p 6 q nq o γ√(p−q) q q p 2 ε npq + max np , p , p < n 6 q < nε , q q 3 γ√(p−q) q p p ε npq + max np , nq , n > q , Mathematics 2021, 9, 501 11 of 32 q q3 nq q o q 2 p γ np + max np , εp , n 6 p < nε 6 q , ( − ) q q 3 γ√p q + max q , p , n q p < nε2, npq np nq 6 p 6 q q q3 q p LR,n(g , ε, γ) = 2 1 γ np + max{εp, 1}, p < n < nε 6 q , ( − ) q 3 γ√p q + max p , 1 , q < n p < nε2, npq nq p 6 q γ√(p−q) p npq + 1, q < n. In particular, 1 LR,n(gC, ∞, · ) ≡ 1, p ∈ 2 , 1 , n ∈ N, for p = 1/2 and all ε, γ > 0, n ∈ N : L (g ) = L (g ) = L (g ) = L (g ) = √1 E,n ∗, ε, γ E,n 0, ε, γ R,n ∗, ε, γ R,n 0, ε, γ min ε, n , LE,n(gC, ε, γ) = LE,n(g1, ε, γ) = LR,n(gC, ε, γ) = LR,n(g1, ε, γ) ≡ 1, for p > 1/2 and all n ∈ N : 2 √q p npq + p, n 6 q , LE,n(g∗, 1, 1) = LE,n(g0, 1, 1) = LE,n(g0, ∞, 1) = 2 2 q√+p p npq , n > q , q2 + p2 LE (g∗, ∞, 1) = √ , ,n npq q q 3 q q p−q p max np , np + p, p , n 6 q , LE (g , ∞, 1) = ,n 0 2 2 q√+p p npq , n > q , LE,n(gC, ε, γ) = 1, ε > 0, γ 6 1. 2 Proof. Observe that in the i.i.d. case under consideration we have Bn = n, 1 g(z) n o L (g ) = √ | (z)| + z 2(z) E,n , ε, γ sup√ γ µ1 σ1 , g( n) 0 2 Let us find µ1( · ), σ1 ( · ). For all z > 0 we have q q 0, z 6 , p q 3 q q | ( )| = E 3 (| | < ) = q q p µ1 z X11 X1 z p , p < z 6 q , q √p−q p pq , z > q , q q 1, z 6 , p 2 2 q q q p σ1 (z) = EX11(|X1| > z) = p, p < z 6 q , q p 0, z > q , Mathematics 2021, 9, 501 12 of 32 q q z, z 6 , p q 3 q q ( ) + 2( ) = q q p γ|µ1 z | zσ1 z γ p + zp, p < z 6 q , q γ(√p−q) p pq , z > q . (1) Compute LE,n(g∗, ε, γ) and LE,n(g0, ε, γ). For all n ∈ N, ε, γ > 0 we have 1 n o L (g ) = √ | (z)| + z 2(z) = E,n ∗, ε, γ sup√ γ µ1 σ1 n 0 2 q ε, nε 6 p , q q q q3 q 2 p = max np , γ np + pε , p < nε 6 q , q q 3 q 3 q q p 1 γ√(p−q) 2 p max np , γ np + nq 1(p > 2 ), npq , nε > q , L (g ) = √1 p = 1 L (g ) > in particular, E,n ∗, ε, γ min ε, n for 2 . Now let us find E,n 0, ε, γ for ε 1. We have √ z ∧ n n o L (g ) = √ | (z)| + z 2(z) = E,n 0, ε, γ sup√ γ µ1 σ1 0 and hence, L (g ) = √1 1(n = ) = √1 > E,n 0, ε, γ max n , 1 n , ε 1, so that we may write for all ε > 0 L (g · ) = √1 = L (g · ) = L (g ∧ · ) E,n 0, ε, min ε, n E,n ∗, ε, E,n ∗, ε 1, . If p > 1/2, then q 3 q γ q + p, z p , n γ 2 o z p 6 q sup |µ1(z)| + σ1 (z) = sup q = √ √ z √ √ γ(√p−q) p n6z<ε n n6z<ε n z pq , z > q , q 3 γ q + p, nε2 p , np 6 q q = q3 γ(p−q) p 2 max γ np + p, p , n 6 q < nε , γ√(p−q) p npq , n > q , Mathematics 2021, 9, 501 13 of 32 q 3 p γ(p−q) γ√(p−q) p and with the account of p 6 nq , p 6 npq for n 6 q ,we finally obtain q q 3 q q + 2 p max np , γ np p , nε 6 q , LE,n(g , ε, γ) = LE,n(g∗, 1, γ) = 0 q q 3 q 3 q q + p γ√(p−q) 2 > p max np , γ np nq , npq , nε q , for all ε > 1 and γ > 0, i.e., for all ε > 0 the identity LE,n(g0, ε, · ) = LE,n(g∗, ε ∧ 1, · ) 1 1 holds also for p > 2 . In particular, with γ = 1 and p > 2 for all ε > 0 we have 2 q ε, nε 6 p , q q 3 q 3 q q + = q + q < 2 p LE,n(g∗, ε, 1) = max np , np pε np pε, p nε 6 q , √ 2 2 2 2 q√+p 2 p max q, q + p , p − q / npq = npq , nε > q , q2 + p2 LE (g∗, ∞, 1) = √ , ,n npq q q3 p np + p, n 6 q , L (g ) = L (g ) = L (g ) = E,n ∗, 1, 1 E,n 0, 1, 1 E,n 0, ∞, 1 2 2 q√+p p npq , n > q . (2) Let us find LE,n(gC, ε, γ) and LE,n(g1, ε, γ). For all n ∈ N, ε, γ > 0 we have q q 1, z 6 , p n γ 2 o γ q q3 q q q p LE (g , ε, γ) = sup |µ (z)| + σ (z) = sup + p, < z , = ,n C √ z 1 1 √ z p p 6 q 0 2 q 1, nε 6 p , q 2 p = max{1, γq + p}, p < nε 6 q , n o 1 γ(p−q) 2 p max 1, (γq + p)1(p > 2 ), p , nε > q , 1 1 in particular, LE,n(gC, ε, γ) ≡ 1 for p = 2 and all ε, γ > 0, as well as for p > 2 , ε > 0, γ 6 1. For ε > 1 we have √ √ z ∨ n n o nL (g ) = | (z)| + z 2(z) = E,n 1, ε, γ sup√ γ µ1 σ1 0 If p = q = 1/2, then n o | (z)| + z 2(z) = z1(z ) = 1(n = ) √ sup √ γ µ1 σ1 √ sup √ 6 1 1 , n6z<ε n n6z<ε n and hence, √ LE,n(g1, ε, γ) = max{1, 1(n = 1)/ n} = 1, ε > 1, γ > 0, n ∈ N; Mathematics 2021, 9, 501 14 of 32 otherwise n o | (z)| + z 2(z) = √ sup √ γ µ1 σ1 n6z<ε n √ 3/2 √ 2 p q q3 q p γq / p + pε n, nε 6 q , γ + zp, z 6 , √ = p q = 2 2 p 2 sup ( − ) q max{γq + p , γ(p − q)}/ pq, n 6 q < nε , √ √ γ√p q , z > p , n6z<ε n pq q √ p γ(p − q)/ pq, n > q , and hence, 3/2 γ√q 2 p max 1, γq + p, np + pε , nε 6 q , √ ( ) = 2 2 p 2 LE,n g1, ε, γ max 1, γq + p, max{γq + p , γ(p − q)}/ npq , n 6 q < nε , γ(p−q) γ√(p−q) γ(p−q) p max 1, γq + p, p , npq = max 1, γq + p, p , n > q , for all ε > 1 and γ > 0. In particular, 3/2 max 1, √q + pε , nε2 p , np 6 q n 2 2 o ( ) = q√+p p 2 > 1 > LE,n g1, ε, 1 max 1, , n 6 < nε , p 2 , ε 1. npq q p 1, n > q , (3) Let us compute LR,n(g∗, ε, γ) and LR,n(g0, ε, γ). With the account of q √ q q ε n, nε2 , z, z 6 p , 6 p nq q √ o q p 2 q q q p p n < n 2 sup zσ (z) = sup pz, < z 6 , = max p , ε , p ε 6 q , √ 1 √ p q 0 q q q q p3 1 q q p3 1 and max p , q 1(p > 2 ) = max p , q , p ∈ [ 2 , 1), for all n ∈ N, ε, γ > 0 we obtain 1 √ L (g ) = √ ( n) + z 2(z) = R,n ∗, ε, γ γ µ1 ε sup√ σ1 n 0 2 q ε, nε 6 p , q q3 nq q o q 2 p = γ np + max np , pε , p < nε 6 q , q q 3 γ√(p−q) + q p 2 > p npq max np , nq , nε q , L (g ) = { √1 } p = 1 > in particular, R,n ∗, ε, γ min ε, n for 2 . Now let ε 1. Taking into account that 2(z) z z 2(z) σ1 does not increase with respect to > 0, and using the just computed sup√ σ1 , 0 1 γ √ √ L (g ) = √ ( n) + {z n} 2(z) = R,n 0, ε, γ µ1 ε sup√ min , σ1 n ε 0 1 γ √ = √ µ (ε n) + sup zσ2(z) = n ε 1 √ 1 0 q 3 γ q + q < 2 p ε np 1, n 6 p nε 6 q , ( − ) γ√p q + n q p < n 2 ε npq 1, 6 p 6 q ε , q 3 γ q nq q o q 2 p = ε np + max np , p , p < n < nε 6 q , nq o γ√(p−q) q q p 2 ε npq + max np , p , p < n 6 q < nε , q q 3 γ√(p−q) q p p ε npq + max np , nq , n > q , p = 1 L (g · ) = √1 = { √1 } = L (g · ) in particular, for 2 we have R,n 0, ε, n min ε, n R,n ∗, ε, for all ε > 1, and hence, for all ε > 0. 2 (4) Let us compute LR,n(gC, ε, γ) and LR,n(g1, ε, γ). Taking into account that σ1 (z) does 2 2 1 not increase with respect to z > 0 and σ1 (0) = σ1 = 1, for all n ∈ N, ε, γ > 0 and p ∈ [ 2 , 1) we obtain γ √ γ √ L (g ) = √ ( n) + 2(z) = √ ( n) + = R,n C, ε, γ µ1 ε sup√ σ1 µ1 ε 1 ε n 0 2 q 1, nε 6 p , q = γ q3 q 2 p ε np + 1, p < nε 6 q , γ√(p−q) 2 p ε npq + 1, nε > q , 1 1 in particular, LR,n(gC, · , · ) ≡ 1 for p = 2 and LR,n(gC, ∞, · ) ≡ 1 for all p ∈ [ 2 , 1). Now let ε > 1. With the account of z 2(z) = √ sup √ σ1 n6z<ε n q nq q o z, z q , q 2 p q 2 6 p max p , p nε ∧ q , n 6 p < nε , q q q = pz, q < z p , = 2 p q p √ sup √ p 6 q p nε ∧ q , p < n 6 q , n z<ε n q 6 p p 0, z > q , 0, n > q , we have √ √ √ nL (g ) = ( n) + (z ∨ n) 2(z) = R,n 1, ε, γ γ µ1 ε sup√ σ1 0 q 3 nq √ √ o q 3 nq √ o q + q = q + q q < 2 p γ p max p , pε n, n γ p max p , pε n , n 6 p nε 6 q , q q 3 √ q q 3 γ(√p−q) q p γ(√p−q) q p q p 2 pq + max p , q , n = pq + max p , q , n 6 p 6 q < nε , q 3 √ √ = q q 2 p γ p + max pε n, n , p < n < nε 6 q , ( − ) q 3 √ γ√p q + max p , n , q < n p < nε2, pq q p 6 q √ γ(√p−q) p pq + n, n > q , 1 in particular, LR,n(g1, ε, γ) ≡ 1 for p = 2 . 3.2. Computation of the Uniform Distance In the present section, we compute the uniform distance ∆1(p) between the d.f. of (27) and the standard normal d.f. Let us denote 1 x2 Ψ(x) = − Φ(−|x|) = Φ(|x|) − , x ∈ R, (33) 1 + x2 1 + x2 Mathematics 2021, 9, 501 16 of 32 s ! s ! 1 − p 1 − p Ψe(p) = Ψ = Φ − (1 − p), 0 < p < 1. (34) p p Please note that Ψ(x) is an even function, therefore, it suffices to investigate it only for x > 0. Lemma 1 (see [27]). The function Ψ(x) is positive for x > 0, increases for 0 < x < xΦ, decreases for x > xΦ and attains its maximal value CΦ = 0.54093 ... in the point xΦ = 0.213105 ... , where xΦ is the unique root of the equation 2 xex /2(1 + x2)−2 = (8π)−1/2. Remark 1. The statement about the interval of monotonicity is absent in the formulation of the corresponding lemma in [27], but these intervals were investigated in the proof. The definition of Ψe and lemma1 immediately imply that the function s ! 1 − p Ψe(p) := Φ − (1 − p) p is positive for p ∈ (0, 1), increases for 0 < p < pΦ, decreases for pΦ < p < 1 and 1 max Ψe(p) = Ψe(pΦ) = CΦ, where pΦ = = 0.9565 . . . . ∈( ) 2 p 0,1 xΦ + 1 Lemma 2. On the interval p ∈ (0, 1) the function r p φ(p) := Ψe(1 − p) + p = Φ 1 − p monotonically increases, while its derivative − p 2(1−p) 0 e φ (p) = √ p 2 2π p(1 − p)3 monotonically decreases. Proof. The function φ(p) monotonically increases as a superposition of monotonically increasing functions. Please note that the derivative φ0(p) takes only positive values, hence, we may define its logarithm p √ 1 3 ln φ0(p) = − − ln(2 2π) − ln p − ln(1 − p) =: u(p). 2(1 − p) 2 2 The derivative 1 1 3 u0(p) = − − + = 0 2(1 − p)2 2p 2(1 − p) changes its sign in the points p ∈ (0, 1) that are the roots of the equation −p − (1 − p)2 + 3p(1 − p) = 0, which has a unique solution p = 1/2 on (0, 1). Since d 8 d 8 u(p)|p= 1 = − , u(p)|p= 3 = − , dp 4 9 dp 4 3 Mathematics 2021, 9, 501 17 of 32 then u(p) is strictly decreasing on (0, 1), and hence, φ0(p) is also strictly decreasing on (0, 1). Theorem 2. Let X1 have distribution (27), then ( p Φ( p/q) − p, 0 < p < 1 , ( ) = P( < ) − ( ) = ( ∨ ) = 2 ∆1 p : sup X1 x Φ x Ψe p q p 1 x∈R Φ( q/p) − q, 2 6 p < 1. Proof. Please note that n p p p p o ∆1(p) = max Φ(− p/q), Φ(− p/q) − q , Φ( q/p) − q , 1 − Φ( q/p) = = max 1 − φ(p), Ψe(q), Ψe(p), 1 − φ(q) = max f1(p), f2(p), f3(p), f4(p) , where f1(p) = 1 − φ(p), f2(p) = Ψe(1 − p) = φ(p) − p, f3(p) = Ψe(p) = φ(1 − p) − (1 − p), f4(p) = 1 − φ(1 − p). Let p > q. Then f1(p) 6 f4(p), and it suffices to show that max f2(p), f4(p) 6 f3(p). Let us prove that f5(p) := f3(p) − f4(p) > 0. Using lemma2, we conclude that the derivative (1−p) − 2p 0 d d e f5(p) := (2φ(1 − p) + p − 2) = 2 φ(1 − p) + 1 = 1 − √ p dp dp 2π (1 − p)p3 decreases on (0, 1) with 0 1 √ 4 0 f5 = 1 − = 0.0321 . . . , lim f5(p) = −∞, 2 2πe p→1−0 hence, f5(p) has a unique stationary point on (0, 1), which is a point of maximum, so that n 1 o 3 inf f5(p) = min f5 , lim f5(p) = min 2Φ(1) − , 0 = min{0.1826 . . . , 0} = 0. 1 2 p→1− 2 2 6p<1 The inequality f2(p) 6 f3(p) follows from the properties of the function Ψe with the account of 1 1 Ψe 2 ∧ Ψe(1 − 0) = Ψe 2 . Thus, the theorem has been proved in the case p > q. The validity of the statement of the theorem for p < q follows from that f1(p) = f4(q) and f2(p) = f3(q). Lemma 3 (see [27,28]). For an arbitrary d.f. F with zero mean and unit variance we have sup|F(x) − Φ(x)| 6 sup Ψ(x) = Ψ(xΦ) = Ψe(pΦ) = ∆1(pΦ) =: CΦ = 0.54093 . . . . x∈R x∈R Remark 2. In [28] [(2.32), (2.33)] it is proved that |F(x) − Φ(x)| 6 Ψ(x), x ∈ R. Mathematics 2021, 9, 501 18 of 32 In [27] it is proved that in the inequality sup|F(x) − Φ(x)| 6 sup Ψ(x) = CΦ x∈R x∈R the equality is attained at a two-point distribution. 4. Lower Bounds for the Exact Constants In the present section, we construct lower bounds for the quantities inf CE(g, ε, γ), inf CR(g, ε, γ), g∈G g∈G CE(ε, γ) = sup CE(g, ε, γ), CR(g, ε, γ) = sup CR(ε, γ) g∈G g∈G and AE(ε, γ) = CE(g∗, ε, γ), AR(ε, γ) = CR(g∗, ε, γ) for all ε, γ > 0. Recall that g0(z) = min{z, Bn}, g1(z) = max{z, Bn}, g∗(z) = z, gC (z) = 1, z > 0. By virtue of invariance of the fractions LE,n(g, ε, γ), LR,n(g, ε, γ) with respect to scale transform of g and extreme property of g1 (see (15)) we have ∆n(F1,..., Fn) ∆n(F1,..., Fn) inf CE(g, ε, γ) = inf sup = inf sup g∈G g∈G L (g, ε, γ) g∈G L (g, ε, γ) F1,...,Fn∈F, E,n F1,...,Fn∈F, E,n n∈N: Bn>0 n∈N: Bn=1 ∆ (F ,..., F ) ∆ (F ,..., F ) sup n 1 n = sup n 1 n = C (g , ε, γ) > L (g , ε, γ) L (g , ε, γ) E 1 F1,...,Fn∈F, E,n 1 F1,...,Fn∈F, E,n 1 n∈N: Bn=1 n∈N: Bn>0 on one hand, and inf CE(g, ε, γ) 6 CE(g1, ε, γ) by definition of the lower bound, on the g∈G other hand. Therefore inf CE(g, ε, γ) = CE(g1, ε, γ), inf CR(g, ε, γ) = CR(g1, ε, γ), g∈G g∈G where the second equality is proved similar to the first one. Let us show that CE(ε, γ) = CE(g0, ε, γ), CR(ε, γ) = CR(g0, ε, γ). Indeed, for arbitrary n ∈ N, F1, ... , Fn ∈ F and g ∈ G due to the extremality of g0 (see (15)) we have ∆n 6 CE(g0, ε, γ)LE,n(g0, ε, γ) 6 CE(g0, ε, γ)LE,n(g, ε, γ), hence, CE(ε, γ) 6 CE(g0, ε, γ), on one hand. On the other hand, this inequality may hold true only with the equality sign, since CE(ε, γ) = sup CE(g, ε, γ) by definition. The same g∈G reasoning holds also true for CR(ε, γ). Thus, CE(g1, ε, γ), CR(g1, ε, γ) are the most optimistic constants, while CE(g0, ε, γ), CR(g0, ε, γ) are the most pessimistic, but universal (exact) ones. The next theorem estab- lishes lower bounds for the exact constants CE(ε, γ) = CE(g0, ε, γ), CE(ε, γ) = CR(g0, ε, γ), and also for the constants AE(ε, γ) and AR(ε, γ) appearing in inequalities (1) and (2). Mathematics 2021, 9, 501 19 of 32 Theorem 3. (i) For all ε, γ > 0 we have Φ(1) − 0.5 0.3413 min{CE(ε, γ), CR(ε, γ), AE(ε, γ), AR(ε, γ)} > . > min{1, ε} min{1, ε} 1 (ii) For 2 < p < 1, q = 1 − p set p ∆1(p) := Φ( q/p) − q. Take any γ > 0. Let us denote for ε 6 1 : ( (p) 1 < p 1 ∆1 /ε, 2 6 ε2+1 , KE,0(p, ε, γ) = p (p) pq p q3 p + p 1 < p < ∆1 max / , γ / ε , ε2+1 1, ( (p) 1 < p 1 ∆1 /ε, 2 6 ε2+1 , KR,0(p, ε, γ) = p (p) q3 p + pq p p 1 < p < ∆1 γ / max / , ε , ε2+1 1, and for ε > 1 : np q o 1 K (p, ε, γ) = ∆ (p)/ max q/p, γ q3/p + p , < p < 1, E,0 1 2 nq q 3 o 2 ∆ (p)/ max q , γ q + εp , ε p < 1, ( ) = 1 p p ε2+1 6 KE,∗ p, ε, γ √ 2 (p) pq q q2 + p2 (p − q) 1 < p < ε ∆1 max , γ , γ , 2 ε2+1 , q . γ q3 nq q o ε2 ∆1(p) + max , p , 2 6 p < 1, ( ) = ε p p ε +1 KR,0 p, ε, γ . ( − ) nq o 2 (p) γ √p q + q p 1 < p < ε ∆1 ε pq max p , , 2 ε2+1 , . q 3 nq o 2 (p) q + q p ε p < ∆1 γ p max p , ε , ε2+1 6 1, KR (p, ε, γ) = ,∗ . ( − ) nq q 3 o 2 (p) γ√p q + q p 1 < p < ε ∆1 pq max p , q , 2 ε2+1 . Then for all ε, γ > 0 we have CE(ε, γ) > sup KE,0(p, ε, γ), CR(ε, γ) > sup KR,0(p, ε, γ), 1 1 2 and for ε > 1 also AE(ε, γ) > sup KE,∗(p, ε, γ), AR(ε, γ) > sup KR,∗(p, ε, γ). 1 1 2 In particular, p √ Φ( q/p) − q pq ( ) > AE ∞, 1 > sup 2 2 0.3703, 1 q + p 2 and for the both constants C• ∈ {CE, CR} with ε = 1, γ = 1 the following lower bound holds: p Φ( q/p) − q min{C•(1, 1), CE(∞, 1)} sup > 0.5685 (p = 0.9058 . . .). > p 3 1 q /p + p 2 Values of the greatest lower bounds of KE,∗(p, ε, γ) and KR,∗(p, ε, γ) for some ε > 1 and γ > 0 are given in Tables1 and2, respectively. Values of the greatest lower bounds of KE,0(p, ε, γ) and KR,0(p, ε, γ) with the corresponding extremal values of p are given in Table4 for some ε > 0 and γ > 0. Mathematics 2021, 9, 501 20 of 32 Table 4. Values of the lower bounds for the constants CE(ε, γ) and CR(ε, γ) obtained in Theorem3 for some ε > 0 and γ > 0. ε γ CE(ε, γ) > p ε γ CR(ε, γ) > p 0.2 0.2 2.78472 0.96368 0.2 0.2 2.78355 0.96415 0.5 0.2 1.17226 0.84124 0.5 0.2 1.17196 0.84770 >1 0.2 0.60108 0.74349 1 0.2 0.60108 0.74349 0.2 0.4 2.76543 0.96318 1.21 0.2 0.60572 0.70768 0.5 0.4 1.14539 0.89145 2 0.2 0.60674 0.80000 >1 0.4 0.58619 0.83356 3 0.2 0.59194 0.68233 0.2 γ∗ 2.74959 0.96276 4 0.2 0.60265 0.68233 0.5 γ∗ 1.13293 0.91254 5.39 0.2 0.61121 0.68233 >1 γ∗ 0.57952 0.86446 8 0.2 0.61947 0.68232 0.2 0.72 2.73337 0.96233 0.2 0.4 2.76388 0.96415 0.5 0.72 1.12389 0.92595 0.5 0.4 1.14539 0.89145 >1 0.72 0.57469 0.88402 1 0.4 0.58619 0.83356 0.2 0.97 2.71102 0.96906 1.76 0.4 0.59815 0.76397 0.5 0.97 1.11352 0.93958 2 0.4 0.59934 0.80000 >1 0.97 0.56910 0.90399 2.63 0.4 0.59228 0.87369 >1 1 0.56854 0.90586 4 0.4 0.57272 0.94118 >1 1.43 0.56207 0.92560 0.2 γ∗ 2.74835 0.96415 >1 1.5 0.56122 0.92795 0.5 γ∗ 1.13293 0.91254 >1 1.62 0.55987 0.93161 1 γ∗ 0.57952 0.86446 >1 2 0.55624 0.94085 1.5 γ∗ 0.58759 0.82642 >1 3 0.54955 0.95569 1.99 γ∗ 0.59346 0.79839 >1 4 0.54507 0.96419 2.12 γ∗ 0.59407 0.81800 >1 5 0.54176 0.96978 3 γ∗ 0.58546 0.90000 >1 ∞ 0 5 γ∗ 0.56465 0.68232 1 Proof. Let n = 1 and the r.v. X1 have the two-point distribution (27) with p ∈ [ 2 , 1). Then, by Theorem2, we have ∆1(F1) = ∆1(p), and all the constants C• ∈ {CE, CE}, A• ∈ {AE, AR} can be bounded from below as ∆1(p) ∆1(p) C•(ε, γ) > sup , A•(ε, γ) > sup . 1 L•,1(g0, ε, γ) 1 L•,1(g∗, ε, γ) 2 6p<1 2 6p<1 1 1 In particular, for p = 2 we have ∆1( 2 ) = Φ(1) − 0.5 = 0.3413 . . . , and by Theorem1 LE,n(g0, ε, γ) = LR,n(g0, ε, γ) = LE,n(g∗, ε, γ) = LR,n(g∗, ε, γ) = min{1, ε}, ε, γ > 0, whence the statement of point (i) follows immediately. 1 To prove point (ii) it suffices to make sure that for all p ∈ ( 2 , 1) and ε, γ > 0 ∆1(p) ∆1(p) K•,0(p, ε, γ) = , K•,∗(p, ε, γ) = , • ∈ {E, R}. L•,1(g0, ε, γ) L•,1(g∗, ε, γ) Mathematics 2021, 9, 501 21 of 32 1 Theorem1 (i) with n = 1 implies that for p > 2 , ε, γ > 0 ε, ε2 q , 6 p q 3 q q q q 2 p LE,1(g∗, ε, γ) = max p , γ p + εp , p < ε 6 q , 2 2 √ 2 p max q, γq + p , γ(p − q) / pq, ε > q . ε, ε2 q , 6 p q 3 nq o q + q q < 2 p LR,1(g∗, ε, γ) = γ p max p , εp , p ε 6 q , nq q 3 o γ(√p−q) q p 2 p pq + max p , q , ε > q , LE,1(g0, ε, γ) = LE,1(g∗, ε ∧ 1, γ), while for ε > 1, by the same Theorem1 (iii), we have q γ q3 nq q o 2 p ε p + max p , p , 1 < ε 6 q , LR,1(g0, ε, γ) = nq o γ(√p−q) q p 2 ε pq + max p , p , 1 6 q < ε , (recall that LR,n(g0, ε, γ) = LR,n(g∗, ε, γ) for ε 6 1). Please note that for all ε > 0 and 1 p ∈ ( 2 , 1) 2 q ⇔ p ∈ 1 1 = > ε 6 p 2 , ε2+1 ∅, ε 1, ( ( 2 p > 1 , p ε > q 2 p ε2+1 > ε2+1 , ε 1, < ε 6 ⇔ 2 ⇔ (35) p q p ε p > 1 > ε2+1 , ε2+1 , ε 6 1, 2 2 > p ⇔ p ∈ 1 ε = ε q 2 , ε2+1 ∅, ε 6 1, 1 so that from the three conditions in (35) under the additional condition p ∈ ( 2 , 1) there remain only two: 2 1 < p 1 ε p < 1 2 6 ε2+1 ε2+1 6 > 1 for ε 6 1, 1 ε2 for ε 1. 2 < p < 1 < p < ε +1 2 ε2+1 In particular, for ε 6 1 1 < p 1 ε, 2 6 ε2+1 , LE,1(g0, ε, γ) = LE,1(g∗, ε, γ) = nq q 3 o q q + p 1 < p < max p , γ p ε , ε2+1 1, 1 < p 1 ε, 2 6 ε2+1 , LR,1(g0, ε, γ) = LR,1(g∗, ε, γ) = q 3 nq o q + q p 1 < p < γ p max p , ε , ε2+1 1, and for ε > 1 nq q 3 o 2 max q , γ q + εp , ε p < 1, ( ) = p p ε2+1 6 LE,1 g∗, ε, γ √ 2 q q2 + p2 (p − q) pq 1 < p ε max , γ , γ / , 2 6 ε2+1 . r nq q q3 o 1 LE,1(g0, ε, γ) = LE,1(g∗, 1, γ) = max p , γ p + p , 2 < p < 1, q q3 nq q o ε2 γ p + max p , εp , 2+ 6 p < 1, L (g , ε, γ) = ε 1 R,1 ∗ ( − ) nq q 3 o 2 γ√p q + q p 1 < p < ε pq max p , q , 2 ε2+1 , Mathematics 2021, 9, 501 22 of 32 q γ q3 nq q o ε2 + max , p , 2 6 p < 1, ( ) = ε p p ε +1 LR,1 g0, ε, γ ( − ) nq o 2 γ √p q + q p 1 < p < ε ε pq max p , , 2 ε2+1 , whence we obtain the above expressions for K•,0, K•,∗. Now let us consider the particular cases. The coinciding lower bounds for CE(∞, 1), p 3 1 CE(1, 1) and CR(1, 1) follow from that LE,1(g0, ∞, 1) = q /p + p for all p ∈ ( 2 , 1), and also from, as it can easily be made sure, q 3 LE,1(g0, 1, 1) = LE,1(g0, ∞, 1) = q /p + p = LR,1(g0, 1, 1) 3 for p > p0, where p0 = 0.6823 ... is the unique root of the equation p + p − 1 = 0 on the segment [0.5, 1]. The lower bound for A (∞, 1) follows from that L (g , ∞, 1) = √ E E,1 ∗ (q2 + p2)/ pq. Now let us find lower bouds for the most optimistic constants CE(g1, ε, γ) and CR(g1, ε, γ). Theorem 4. (i) For all ε, γ > 0 we have min{CE(g1, ε, γ), CR(g1, ε, γ)} > Φ(1) − 0.5 > 0.3413. 1 (ii) For 2 < p < 1, q = 1 − p set p ∆1(p) := Φ( q/p) − q. Take any γ > 0. Let us denote for ε 6 1 : ( (p) 1 < p 1 ∆1 , 2 6 ε2+1 , KE,1(p, ε, γ) = (p) { q + p} 1 < p < ∆1 / max 1, γ , ε2+1 1, (p) 1 < p 1 ∆1 , 2 6 ε2+1 , KR,1(p, ε, γ) = q 3 (p) + γ q 1 < p < ∆1 / 1 ε p , ε2+1 1, and for ε > 1 : n 2+ 2 ( − ) o 2 (p) q + p γq√ p γ√p q 1 < p < ε ∆1 / max 1, γ , pq , pq , 2 ε2+1 , KE,1(p, ε, γ) = 3/2 2 (p) q + p γ√q + p ε p < ∆1 / max 1, γ , p ε , ε2+1 6 1, q 3 2 γ(√p−q) p 1 ε ∆1(p)/ + max , 1 , < p < 2 , K (p ) = pq q 2 ε +1 R,1 , ε, γ q 3 2 (p) q + { p } ε p < ∆1 / γ p max ε , 1 , ε2+1 6 1. Then for all ε, γ > 0 we have CE(g1, ε, γ) > sup KE,1(p, ε, γ), CR(g1, ε, γ) > sup KR,1(p, ε, γ). 1 1 2 In particular, CE(g1, ε, γ) > sup ∆1(p) = ∆1(pΦ) =: CΦ = 0.5409 . . . , 1 2 2 −1 if γ 6 1, ε 6 1 or ε 6 xΦ, where xΦ = 0.213105 . . . is defined in Lemma 1, pΦ = (xΦ + 1) = 0.9565 . . . , p √ Φ( q/p) − q pq ( ) > CE g1, ∞, 1 > sup 2 2 0.3703, 1 q + p 2 CR(g1, ε, γ) > CΦ for ε 6 xΦ, ∆1(p) CR(g , 1, 1) sup KR (p, 1, 1) = > 0.5370. 1 > ,1 p 3 1 1 + (1 − p) /p p=0.9678... 2 Values of the greatest lower bounds of KE,1(p, ε, γ) and KR,1(p, ε, γ) with the corre- sponding extremal values of p are given in Table5 for some ε > 0 and γ > 0. Table 5. Values of the lower bounds for the most optimistic constants CE(g1, ε, γ) and CR(g1, ε, γ) obtained in Theorem3 for some ε > 0 and γ > 0. ε γ CE(g1, ε, γ) > p ε γ CR(g1, ε, γ) > p 0.2 0.2 0.54093 0.95655 0.2 0.2 2.78355 0.96415 0.5 0.2 0.54093 0.95655 0.5 0.2 1.17196 0.84770 >1 0.2 0.54093 0.95655 1 0.2 0.60108 0.74349 0.2 0.4 0.54093 0.95655 1.21 0.2 0.60572 0.70768 0.5 0.4 0.54093 0.95655 2 0.2 0.60674 0.80000 >1 0.4 0.54093 0.95655 3 0.2 0.59194 0.68233 0.2 γ∗ 0.54093 0.95655 4 0.2 0.60265 0.68233 0.5 γ∗ 0.54093 0.95655 5.39 0.2 0.61121 0.68233 >1 γ∗ 0.54093 0.95655 8 0.2 0.61947 0.68232 0.2 0.72 0.54093 0.95655 0.2 0.4 2.76388 0.96415 0.5 0.72 0.54093 0.95655 0.5 0.4 1.14539 0.89145 >1 0.72 0.54093 0.95655 1 0.4 0.58619 0.83356 0.2 0.97 0.54093 0.95655 1.76 0.4 0.59815 0.76397 0.5 0.97 0.54093 0.95655 2 0.4 0.59934 0.80000 >1 0.97 0.54093 0.95655 2.63 0.4 0.59228 0.87369 >1 1 0.54093 0.95655 4 0.4 0.57272 0.94118 >1 1.43 0.53297 0.97214 0.2 γ∗ 2.74835 0.96415 >1 1.5 0.53197 0.97383 0.5 γ∗ 1.13293 0.91254 >1 1.62 0.53041 0.97637 1 γ∗ 0.57952 0.86446 >1 2 0.52637 0.98230 1.5 γ∗ 0.58759 0.82642 >1 3 0.51963 0.99024 1.99 γ∗ 0.59346 0.79839 >1 4 0.51568 0.99379 2.12 γ∗ 0.59407 0.81800 >1 5 0.51307 0.99569 3 γ∗ 0.58546 0.90000 >1 ∞ 0 5 γ∗ 0.56465 0.68232 Remark 3. Theorem4 yields the following lower bound for the exact constant AE(∞, 1) in the Esseen-type inequality (1): p √ Φ( q/p) − q pq ( ) ( ) > AE ∞, 1 > CE g1, ∞, 1 > sup 2 2 0.3703, 1 q + p 2 which coincides with the one obtained directly in Theorem3. Mathematics 2021, 9, 501 24 of 32 1 Proof. Let n = 1 and the r.v. X1 have the two-point distribution (27) with p ∈ [ 2 , 1). Then, by Theorem2 we have ∆1(F1) = ∆1(p), and all the constants C• ∈ {CE, CE} are bounded from below as ∆1(p) C•(g1, ε, γ) > sup . 1 L•,1(g1, ε, γ) 2 6p<1 1 1 In particular, for p = 2 we have ∆1( 2 ) = Φ(1) − 0.5 = 0.3413 . . . , while by Theorem1 LE,n(g1, ε, γ) = LR,n(g1, ε, γ) ≡ 1, ε, γ > 0, whence the statement of point (i) follows immediately. 1 To prove point (ii), it suffices to make sure that for all p ∈ ( 2 , 1), ε, γ > 0 we have ∆1(p) K•,1(p, ε, γ) = , • ∈ {E, R}. L•,1(g1, ε, γ) 1 Theorem1 (i) (see (30) and (32)) with n = 1 implies that for p > 2 , γ > 0 and ε 6 1 we have 2 q 1 1, ε 6 ⇔ p 6 2 , ∆ (p) L (g , ε, γ) = L (g , ε, γ) = p ε +1 = 1 , E,1 1 E,1 C 2 q 1 { + } > ⇔ > KE,1(p, ε, γ) max 1, γq p , ε p p ε2+1 , q 1, ε2 6 , ∆ (p) L (g , ε, γ) = L (g , ε, γ) = p = 1 , R,1 1 R,1 C γ q q3 q 2 KR,1(p, ε, γ) ε p + 1, ε > p , while for ε > 1, by Theorem1 (ii) and (iii), respectively, we have 3/2 2 + γ√q + 2 p ⇔ ε max 1, γq p, p pε , ε 6 q p > ε2+1 , ∆1(p) L (g , ε, γ) = 2 2 = , E,1 1 γq +p γ(p−q) 2 p ε2 + √ √ > ⇔ < KE,1(p, ε, γ) max 1, γq p, pq , pq , ε q p ε2+1 , q 3 γ q + max{εp, 1}, ε2 p , p 6 q ∆1(p) L (g , ε, γ) = q 3 = , R,1 1 γ(p−q) p 2 p √ + > KR,1(p, ε, γ) pq max q , 1 , ε q , which coincides with the statement of point (ii). Now let us consider the particular cases. The lower bound for CE(g1, ε, γ) with γ 6 1 1 follows from that, for the specified γ and ε 6 1, we have KE,1(p, ε, γ) = ∆1(p), 2 < p 6 1. 1 1 For ε 6 xΦ we have 2+ > 2 = pΦ and, hence, ε 1 xΦ+1 CE(g1, ε, γ) > sup KE,1(p, ε, γ) = sup ∆1(p) = ∆1(pΦ) = CΦ. 1 The lower bound for CR(g1, ε, γ) > CΦ with ε 6 xΦ is obtained similarly. The given values of the constants CE(g1, ∞, 1), CR(g1, 1, 1) are computed trivially. 5. Lower Bounds for the Asymptotically Best Constants Let us investigate the constants CAB(g, ε, γ) in (11) and (12) and construct their lower bounds. Due to the extremity of the functions g0(z) = min{z, Bn} and g1(z) = max{z, Bn} (see (15)) we have sup CAB(g, ε, γ) = CAB(g0, ε, γ), inf CAB(g, ε, γ) = CAB(g1, ε, γ). g∈G g∈G in both inequalities (11) and (12). Mathematics 2021, 9, 501 25 of 32 Theorem 5. For all ε, γ > 0 : (i) in inequality (11) 1 p + 1 ( ) √ = ( ) inf CAB g0, ε, γ > sup 2 2 : CAB γ , (36) ε>0 3 2π 1 max{1 − p, γ(1 − p) + p , γ(2p − 1)} 2 (iii) in both inequalities (11) and (12) CAB(g1, ε, γ) = 0. Values of CAB(γ) for some γ and the corresponding extreme values of p are given in Table6. Table 6. Values ofthe lower bound CAB(γ) (see (36)), rounded down, for the asymptotically best constant CAB(g0, ε, γ) from inequality (11) for some γ. The second line contains rounded extreme values of p in (36). γ 0.1 0.2 0.4 0.56 1 1.5 2 3 4 5 p 0.6112 0.6039 0.5871 0.5710 0.5812 0.6733 0.6666 0.6340 0.6202 0.6126 CAB(γ) 0.5511 0.5384 0.5111 0.4868 0.4097 0.3627 0.3324 0.2703 0.2240 0.1904 Proof. Let us show that CAB(g1, ε, γ) = 0. According to (13) and (14), LE,n(g1, ε, γ) > 1, LR,n(g1, ε, γ) > 1 for all ε, γ > 0, so that for the both inequalities (11) and (12) we have CAB(g1, ε, γ) 6 sup lim sup ∆n(F) = 0, whence, with the account of non-negativity of F n→∞ CAB(g1, ε, γ), the statement of point (iii) follows. Now let us estimate from below the constants CAB(g0, ε, γ). Take i.i.d. r.v.’s X1, ... , Xn 2 p with distribution (27), where p > 1/2. Then, by virtue of Theorem1, for n(ε ∧ 1) > q we have max{q, γq2 + p2, γ(p − q)} L (g , ε, γ) = √ . E,n 0 npq √ The r.v. X1 is lattice with a span h = 1/ pq. The Esseen asymptotic expansion [18] for lattice distributions with the span h implies that √ EX3 + 3hσ2 lim sup ∆ n = √1 1 . n 3 n→∞ 6 2πσ1 2 3 √ For the chosen distribution of X1 we have σ1 = 1, EX1 = (p − q)/ pq, therefore √ p − q + 3 p + 1 lim ∆n n = p = p , n→+∞ 6 2πpq 3 2πpq and, hence, in inequality (11) we have √ ∆ ∆n npq C (g , ε, γ) lim n = lim = AB 0 > → → 2 2 n ∞ LE,n(g0, ε, γ) n ∞ max{q, γq + p , γ(p − q)} Mathematics 2021, 9, 501 26 of 32 p + 1 = √ 3 2π · max{q, γq2 + p2, γ(p − q)} for all p ∈ (1/2, 1), ε, γ > 0, whence, with the account of arbitrariness of the choice of p, the statement of point (ii) follows. In particular, for γ = 1 we obtain √ p + 1 1 EX3 + 3hσ2 10 + 3 C (g , ε, 1) sup √ = √ sup 1 1 = √ = 0.4097 . . . AB 0 > 2 2 3 1 3 2π(p + q ) 6 2π 1 E|X1| 6 2π 2 and denoting γ a = γ(ε−1 ∧ 1) = , ε ∨ 1 we obtain √ p + 1 1 5 − 1 , p ∈ , , √ ∆ a(2p − 1) + 1 − p 2 2 f (p) := 3 2π lim n = √ n→∞ LR,n(g0, ε, γ) p + 1 5 − 1 , p ∈ , 1 . a(2p − 1) + p2 2 √ 1 5−1 Let us investigate the behaviour of f (p) in dependence of a > 0. For p ∈ 2 , 2 the numerator of f 0(p) takes the form a(2p − 1) + 1 − p − (p + 1)(2a − 1) = 2 − 3a, f (p) a > 2 a < 2 so that is monotonically√ decreasing, if 3 , and monotonically increasing, if 3 , 5−1 0 while for p ∈ 2 , 1 the numerator of f (p) has the form a(2p − 1) + p2 − 2(p + 1)(a + p) = −p2 − 2p − 3a < 0, 1/2 < p < 1, and hence, f (p) decreases strictly monotonically. Therefore, √ √ 5 − 1 1 + 5 √ f = √ √ , a < 2 , 2 3 3 2π · CAB(g0, ε, γ) > sup f (p) = 2a( 5 − 2) + 3 − 5 1/2 whence the statement of point (ii) follows immediately. 6. Lower Bounds for the Asymptotically Exact Constants First of all, please note that asymptotically exact constants CAE(g1, ε, γ) and the lower asymptotically exact constants CAE(g1, ε, γ) are defined for none of the inequalities (11), (12), since the corresponding fractions LE,n(g1, ε, γ), LR,n(g1, ε, γ) are bounded from below by one uniformly with respect to ε and γ (see (13) and (14)) and, hence, cannot be infinitesimal. Theorem 6. In the both inequalities (11), (12) for all ε, γ > 0 we have 1 CAE(g0, ε, γ) > √ , (38) 2 2π Mathematics 2021, 9, 501 27 of 32 ∗ 1 sup CAE(g, ε, γ) = CAE(g0, ε, γ) > CAE(g0, ε, γ) > , g∈G 2(ε ∧ 1) 0.5, ε 6 1, ∗ −2 −2 inf CAE(g, ε, γ) = CAE(g1, ε, γ) > CAE(g1, ε, γ) > exp{−ε }I0(ε )/2, 1 < ε 1.1251, g∈G 6 0.2344, ε > 1.1251. ∞ 2k 2 where I0(z) = ∑k=0 (z/2) /(k!) is the modified Bessel function of the zero order. As a lower bound to the asymptotically exact constant CAE(g0, ε, γ), due to (26), both the asymptotically best CAB(g0, ε, γ) and the lower asymptotically exact CAE(g0, ε, γ) con- stants√ may serve. However, the lower bound of the lower asymptotically exact constant (2 2π)−1 = 0.1994 ... stated in (38) turns to be less accurate in Rozovskii-type inequality, than the lower bound of the asymptotically best constant in (37), since the minorant in (37) is monotone with√ respect to γ for ε 6 1 and with respect to γ/ε for ε > 1 varying within −1 2 the range from ( 2π) as γ/(ε ∨ 1) → 3 to √ 1 + 5 γ √ √ = 0.5633 . . . as → 0 3 2π 3 − 5 ε ∨ 1 √ and thus staying always greater than (2 2π)−1. The minorant to the asymptotically best constant in (36) monotonically decreases with respect to γ and√ does not depend on ε. Hence, > C ( ) = ( )−1 > there exists a unique value γ0 0 such√ that AB √γ0 2 2π (and even√ γ0 1 due = C ( ) = ( + ) ( ) = > ( )−1 to that for γ 1 we have AB 1 √10 3 / 6 2π 0.4097 ... 2 2π ), so that −1 for all γ < γ0 we have CAB(γ) > (2 2π) . It is easy to make sure that γ0 = 4.7010 ... Therefore, as a lower bound for the constant CAE(g0, ε, γ) in Esseen-type inequality (11) it is reasonable to choose the lower bound in (38) for γ 6 γ0 and the lower bound in (36) for γ > γ0. Let us formulate this as a corollary. Corollary 1. For the asymptotically exact constant in inequality (11) the lower bound inf CAE(g0, ε, γ) > CAB(γ ∧ γ0), ε>0 ( ) = holds, where√CAB γ is defined in (36), and γ0 4.7010 ... is the unique root of the equation −1 CAB(γ) = (2 2π) for γ > 0. In particular, with the account of Table3 the following two-sided bounds hold: √ √10+3 0.4097 . . . = CAE(g , 1, 1) 1.80. 6 2π 6 0 6 For the asymptotically exact constant in (12) we have √ 1 + 5 γ 2 √ √ √ , < , 3 2π 2γ(ε−1 ∧ 1)( 5 − 2) + 3 − 5 ε ∨ 1 3 CAE(g0, ε, γ) > 1 γ 2 √ , > . 2π ε ∨ 1 3 In particular, with the account of Table3 the following two-sided bounds hold: √1 0.3989 . . . = CAE(g , 1, 1) 1.80. 2π 6 0 6 Proof of Theorem6. The relations between the constants follow from their definitions ∗ (see also (26)). Therefore, it remains to prove the lower bounds for CAE(g0, ε, γ), CAE(g0, ε, γ), ∗ and CAE(g1, ε, γ). Following [29] ([§ 2.3.2]), consider i.i.d. r.v.’s X1, ... , Xn with a symmetric tree-point distribution P(|X1| = 1) = p = 1 − P(X1 = 0) ∈ (0, 1), Mathematics 2021, 9, 501 28 of 32 whose d.f. will be denoted by Fp(x) = P(X1 < x), x ∈ R. Then ( 2 2 2 p, z 6 1, EX1 = 0, EX = p, σ (z) = EX 1(|X1| > z) = µ1( · ) ≡ 0, 1 1 1 0, z > 1, 2 √ √ Bn = np, g0(z) = min{z, np}, g1(z) = max{z, np}, and the fractions LE,n, LR,n coincide, do not depend on γ and take the form g(z) n g(z)σ2(z) L (g, ε) := L (g, ε, · ) = L (g, ε, · ) = sup σ2(z) = sup √1 . n E,n R,n 2 ( ) ∑ k √ pg( np) 0 Due to the monotonicity of g ∈ G, we have √ √ g(z) 2(z) = pg( np ∧ ) = p {g( ) g( np)} sup√ σ1 ε 1 min 1 , ε , 0 and hence, √ ming(1), g(ε np) Ln(g, ε) = √ , ε > 0, n ∈ N, p ∈ (0, 1), g( np) in particular, 1 1 L (g , ε) = min √ , ε, 1 , L (g , ε) = min √ ∨ 1, ε ∨ 1 , ε > 0, n 0 np n 1 np −2 −2 and Ln(g1, ε) = 1 for ε 6 1. Moreover, for all n > ` ∨ ε ∨ 1 we have n n 1 o o {p ∈ (0, 1) : L (g , ε) = `} = p ∈ (0, 1) : min √ , ε, 1 = ` = n 0 np `−2/n =: p(`) , ` < ε ∧ 1, = 0, (ε−2 ∨ 1)/n, ` = ε ∧ 1, for all ε > 0, ∅, ` > ε ∧ 1, (0, 1), ` = 1, {p : ` = 1} = ε 6 1, ∅, otherwise, {p ∈ (0, 1) : Ln(g1, ε) = `} = n o p(`), ` ∈ [1, ε], √1 ∨ = ` = > p : min ε, np 1 ε 1. ∅, otherwise, As a result, the fractions do not depend on γ, we obtain the lower bounds inf CAE(g0, ε, γ) > lim sup lim sup sup ∆n(Fp)/` = lim sup lim sup ∆n(Fp(`))/`, γ>0 → → `→0 n ∞ p∈(0,1) : Ln(g0,ε)=` `→0 n ∞ ∗ inf CAE(g, ε, γ) > sup lim sup sup ∆n(Fp)/`, ε > 0, γ>0 → `>0 n ∞ p∈(0,1) : Ln(g,ε)=` in particular, for all γ > 0 we have ∗ n o CAE(g0, ε, γ) > max sup lim sup ∆n(Fp(`))/`, lim sup sup ∆n(Fp)/(ε ∧ 1) , 0<`<ε∧1 n→∞ n→∞ 0 Let us find the lower bound for the uniform distance ∆n(Fp). Due to the symmetry, we have 2P(Sn < 0) + P(Sn = 0) = 1, i.e., P(Sn < 0) = (1 − P(Sn = 0))/2, whence for even n we obtain P(S = 0) (1 − p)n n/2 n! p/2 2k ∆ (F ) Φ(0) − P(S < 0) = n = . n p > n ∑ ( − ) ( )2 − 2 2 k=0 n 2k ! k! 1 p 1 Please note that lim sup ∆n(Fp) > 2 , while with p = α/n, α ∈ (0, n) we have p→0 1 α n n/2 n! 1/2 2k e−α + δ (α) n/2 ∆ (F ) 1 − = n u (α), n α/n > ∑ ( − ) ( )2 − ∑ k,n 2 n k=0 n 2k ! k! n/α 1 2 k=0 where n! 1/2 2k uk,n(α) = , lim δn(α) = 0, α > 0. (n − 2k)!(k!)2 n/α − 1 n→∞ In [29] ([pp. 268–269]) it was shown that for every α > 0 n/2 ∞ (α/2)2k lim sup u (α) = I (α). ∑ k,n > ∑ ( )2 0 n→∞ k=0 k=0 k! Therefore, 1 −α lim sup ∆n(Fα/n) > 2 e I0(α), ` > 0. (39) n→∞ Let us bound from below expressions such as supp ∆n(Fp) as 1 sup ∆n(Fp) > lim sup ∆n(Fp) > 2 . (40) p∈(0, · ) p→0 From (39) with α = `−2 we obtain √ 1 −α 1 CAE(g0, ε, γ) lim sup lim sup ∆n(F (`))/` lim αe I0(α) = √ , ε, γ > 0. > p > 2 α→∞ `→0 n→∞ 2 2π Inequalities (39) and (40) imply that ∗ n o CAE(g0, ε, γ) > max sup lim sup ∆n(Fp(`))/`, lim sup sup ∆n(Fp)/(ε ∧ 1) > 0<`<ε∧1 n→∞ n→∞ 0 −1 sup f (α) > f (1) = e I0(1) = 0.4657 . . . , α>ε−2∨1 −2 √ f (α∗) > 0.4688, ε < α∗, ⇔ ε > 1/ α∗ = 1.1250 . . . , sup f (α) > −2 −2 −2 −2 √ ε−2<α<1 f (ε ) = exp{−ε }I0(ε ), ε > α∗, ⇔ ε 6 1/ α∗. Mathematics 2021, 9, 501 30 of 32 Hence, we finally obtain ∗ ( ) 1 { −1 } = 1 ( −1 ∨ ) CAE g0, ε, γ > 2 max 0.4657, ε , 1 2 ε 1 , ( 1 exp{−ε−2}I (ε−2), 1 < ε 1.1251, ∗ ( ) 2 0 6 CAE g1, ε, γ > 1 2 · 0.4688 = 0.2344, ε > 1.1251. 7. Conclusions In the present work, we introduced a detailed classification of the exact and asymptoti- cally exact constants in natural convergence rate estimates in the Lindeberg’s theorem such as Esseen’s and Rozovskii’s inequalities. We found the lower bounds for the exact (univer- sal) constants and the most optimistic absolute constants. Also we constructed the lower bounds for the asymptotically best, the lower asymptotically exact and the conditional upper asymptotically exact constants. With the account of the previously known upper estimates for the asymptotically exact constants, this allowed to obtain two-sided bounds for all of the introduced asymptotic constants. The values of the constructed bounds were computed for some ε, γ > 0 and the results of the calculations were presented in Tables1–5 . As we can see, the obtained lower bounds are note far from the upper ones, so that, from the point of practical use, there is no high motivation to improve the method of construction of the upper bounds. As regards the historical values ε = γ = 1 in both inequalities (11) and (12), ε = ∞, γ = 1 in (11), we obtained the following results for the absolute constants: p Φ( (1 − p)/p) − 1 + p min{CE(1, 1), CR(1, 1), CE(∞, 1)} > 0.5685. > p 3 (1 − p) /p + p p=0.9058... Recall that CE(∞, 1) 6 CE(1, 1) = AE(1, 1) 6 2.73, CR(1, 1) = AR(1, 1) 6 2.73 (see Tables1 and2), and, as it follows from the presented results, these upper bounds cannot be lowered more than by 5 times. As regards the most optimistic constants, i.e., the values of the constants under the “best” choice of the function g ∈ G, we have shown that they cannot nevertheless be less than p Φ( (1 − p)/p) − 1 + p CR(g , 1, 1) > 0.5370. 1 > p 3 1 + (1 − p) /p p=0.9678... CE(g1, 1, 1) > CΦ = 0.5409 . . . , p p Φ( (1 − p)/p) − 1 + p p(1 − p) ( ) > CE g1, ∞, 1 > 2 2 0.3703. (1 − p) + p p=0.6090... Since AE(∞, 1) > CE(g1, ∞, 1), the latest inequality immediately yields the lower bound AE(∞, 1) > 0.3703 that was also obtained directly in Theorem3. Recall that AE(∞, 1) 6 2.66 (see Table1). Asymptotic lower bounds for CE and CR for infinitely large sample sizes n may be constructed in terms of the upper asymptotically exact CAE and the conditional upper ∗ ∗ asymptotically exact CAE constants which are linked as CAE 6 CAE. We constructed the lower ∗ bounds for CAE(g, ε, γ) with g = g0, g1 which turned out to coincide for ε = γ = 1: ∗ ∗ CAE(g0, 1, 1) > CAE(g1, 1, 1) > 0.5 Mathematics 2021, 9, 501 31 of 32 in both inequalities (11) and (12) (the same lower bound 0.5 was obtained directly for ∗ CAE(g0, 1, 1) in Theorem6). The next series of the lower bounds for CE, CR was obtained under the additional assumptions on the smallness of the fractions LE,n, LR,n, which allow constructing more opti- mistic upper bounds, for example, in terms of the asymptotically exact constants, i.e., to con- sider estimates such as (25) with C∗ = CAE. Recall that in this situation the following upper AE AE bounds are known from [11]: AE (1, 1) 6 1.80, AE (1, 1) 6 1.80. Observation that the con- AE AE stants AE (1, 1), AR (1, 1) coincide with CAE(g0, 1, 1) in (22) with Ln = LE,n, LR,n, respectively, AE AE allows taking as a lower bound to AE (1, 1), AE (1, 1) (or, more generally, to CAE(g0, 1, 1)) C (g ) C (g ) C (g ) {C (g ) C (g )} any of AB 0, 1, 1 and AE 0, 1, 1 : AE √0, 1, 1 > max AB 0, 1, 1 , AE 0, 1, 1 . How- C (g ) ( )−1 = ever, the lower bound AE 0, 1, 1 > 2 2π 0.1994 ...√in the both inequalities (11) √10+3 and (12) is less than each of the lower bounds CAB(g0, 1, 1) > = 0.4097 ... in (11) and √ 6 2π CAB(g0, 1, 1) > 1/ 2π = 0.3989 ... in (12), so that the best choice of the lower bound for CAE(g0, 1, 1) is CAB(g0, 1, 1) in both inequalities (11) and (12). Hence, the following two-sided bounds hold true: AE AE 0.4097 < AE (1, 1) 6 1.80, 0.3989 < AE (1, 1) 6 1.80. AE AE Therefore, the asymptotic values AE (1, 1), AR (1, 1) of the constants CE(1, 1) and CR(1, 1) in (11), (12), valid for small LE,n, LR,n, can neither be improved more than by 4.5 times. Further research might include construction of the estimates of the rate of convergence in non-classical Lindeberg’s theorem whose conditions were obtained in a recent paper [30] or some extensions to the case of dependent random summands (see [31]). Author Contributions: Conceptualization, I.S.; methodology, I.S.; formal analysis, I.S., R.G., V.M.; investigation, I.S., R.G., V.M.; writing—original draft preparation, V.M.; writing—review and editing, I.S.; supervision, I.S.; funding acquisition, I.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Russian Foundation for Basic Research (projects Nos. 19-07-01220-a, 20-31-70054) and by the Ministry for Education and Science of Russia (grant No. MD–5748.2021.1.1). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors would like to thank an anonymous referee for pointing out the reference to [31]. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations The following abbreviations are used in this manuscript: r.v. random variable i.i.d. independent identically distributed d.f. distribution function w.r.t. with respect to References 1. Berry, A.C. The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 1941, 49, 122–136. [CrossRef] 2. Esseen, C.G. On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astron. Fys. 1942, A28, 1–19. 3. Osipov, L.V. Refinement of Lindeberg’s theorem. Theory Probab. Appl. 1966, 11, 299–302. [CrossRef] 4. Petrov, V.V. An estimate of the deviation of the distribution function of a sum of independent random variables from the normal law. Sov. Math. Dokl. 1965, 6, 242–244. Mathematics 2021, 9, 501 32 of 32 5. Esseen, C.G. On the remainder term in the central limit theorem. Arkiv för Matematik 1969, 8, 7–15. [CrossRef] 6. Rozovskii, L.V. On the rate of convergence in the Lindeberg–Feller theorem. Bull. Leningr. Univ. 1974, 1, 70–75. (In Russian) 7. Wang, N.; Ahmad, I.A. A Berry–Esseen inequality without higher order moments. Sankhya A Indian J. Stat. 2016, 78, 180–187. [CrossRef] 8. Gabdullin, R.; Makarenko, V.; Shevtsova, I. Esseen–Rozovskii type estimates for the rate of convergence in the Lindeberg theorem. J. Math. Sci. 2018, 234, 847–885. [CrossRef] 9. Gabdullin, R.; Makarenko, V.; Shevtsova, I. A generalization of the Wang–Ahmad inequality. J. Math. Sci. 2019, 237, 646–651. [CrossRef] 10. Gabdullin, R.; Makarenko, V.; Shevtsova, I. A generalization of the Rozovskii inequality. J. Math. Sci. 2019, 237, 775–781. [CrossRef] 11. Gabdullin, R.; Makarenko, V.; Shevtsova, I. On natural convergence rate estimates in the Lindeberg theorem. Indian J. Statis. Sankhya A 2020.[CrossRef] 12. Korolev, V.; Dorofeyeva, A. Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions. Lith. Math. J. 2017, 57, 38–58. [CrossRef] 13. Zolotarev, V.M. Modern Theory of Summation of Random Variables; VSP: Utrecht, The Netherlands, 1997. 14. Ibragimov, I.A. On the accuracy of the approximation of distribution functions of sums of independent variables by the normal distribution. Theory Probab. Appl. 1966, 11, 559–579. [CrossRef] 15. Katz, M.L. Note on the Berry–Esseen theorem. Ann. Math. Stat. 1963, 34, 1107–1108. [CrossRef] 16. Korolev, V.Y.; Popov, S.V. Improvement of convergence rate estimates in the central limit theorem under weakened moment conditions. Dokl. Math. 2012, 86, 506–511. [CrossRef] 17. Kolmogorov, A.N. Some recent works in the field of limit theorems of probability theory. Bull. Mosc. Univ. 1953, 10, 29–38. (In Russian) 18. Esseen, C.G. A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr. 1956, 39, 160–170. [CrossRef] 19. Chistyakov, G.P. Asymptotically proper constants in the Lyapunov theorem. J. Math. Sci. 1999, 93, 480–483. [CrossRef] 20. Chistyakov, G.P. A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. I. Theory Probab. Appl. 2002, 46, 226–242. Original Russian text: Teor. Veroyatn. Primen. 2001, 46, 326–344. [CrossRef] 21. Chistyakov, G.P. A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. II. Theory Probab. Appl. 2002, 46, 516–522. Original Russian text: Teor. Veroyatn. Primen. 2001, 46, 573–579. [CrossRef] 22. Chistyakov, G.P. A new asymptotic expansion and asymptotically best constants in Lyapunov’s theorem. III. Theory Probab. Appl. 2003, 47, 395–414. Original Russian text: Teor. Veroyatn. Primen. 2002, 46, 475–497. [CrossRef] 23. Shevtsova, I.G. The lower asymptotically exact constant in the central limit theorem. Dokl. Math. 2010, 81, 83–86. [CrossRef] 24. Shevtsova, I.G. On the asymptotically exact constants in the Berry–Esseen–Katz inequality. Theory Probab. Appl. 2011, 55, 225–252. Original Russian text: Teor. Veroyatn. Primen. 2010, 55, 271–304. 25. Shevtsova, I.G. On the absolute constants in the Berry–Esseen-type inequalities. Dokl. Math. 2014, 89, 378–381. [CrossRef] 26. Ibragimov, I.A.; Linnik, Y.V. Independent and Stationary Sequences of Random Variables; Wolters–Noordhoff Publishing: Groningen, The Netherlands, 1971. 27. Chebotarev, V.I.; Kondrik, A.S.; Mikhailov, K.V. On an Extreme Two-Point Distribution. arXiv 2007, arXiv:0710.3456. 28. Agnew, R.P. Estimates for global central limit theorems. Ann. Math. Stat. 1957, 28, 26–42. [CrossRef] 29. Shevtsova, I.G. Accuracy of the Normal Approximation: Methods of Estimation and New Results; Argamak–Media: Moscow, Russia, 2016. (In Russian) 30. Ibragimov, I.A.; Presman, E.L.; Formanov, S.K. On modifications of the Lindeberg and Rotar’ conditions in the central limit theorem. Theory Probab. Appl. 2020, 65, 648–651. [CrossRef] 31. Gómez-García, J.G.; Chesneau, C. A dependent Lindeberg central limit theorem for cluster functionals on stationary random fields. Mathematics 2021, 9, 212. [CrossRef]